We obtain a uniform linear bound for the Chevalley function at a point in
the source of an analytic mapping that is regular in the sense of
Gabrielov. There is a version of
Chevalley's lemma also along a fibre, or at a point of the image of a proper
analytic mapping. We get a uniform linear bound for the Chevalley
function of a closed Nash (or formally Nash) subanalytic set.

We obtain a uniform linear bound for the Chevalley function at a point in
the source of an analytic mapping that is regular in the sense of
Gabrielov. There is a version of
Chevalley's lemma also along a fibre, or at a point of the image of a proper
analytic mapping. We get a uniform linear bound for the Chevalley
function of a closed Nash (or formally Nash) subanalytic set.

We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our j-functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using j, we construct the fields of moduli and definition
for some moduli problems associated to the AtkinLehner group
actions.

We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our $j$-functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using $j$, we construct the fields of moduli and definition
for some moduli problems associated to the Atkin--Lehner group
actions.

This paper is a continuation of three recent articles
concerning the structure of hyperinvariant
subspace lattices of operators on a (separable, infinite dimensional)
Hilbert space H. We show herein, in particular, that
there exists a "universal" fixed block-diagonal operator B on
H such that if ε > 0 is given and T is
an arbitrary nonalgebraic operator on H, then there exists
a compact operator K of norm less than ε such that
(i) Hlat(T) is isomorphic as a complete lattice to Hlat(B + K)
and (ii) B + K is a quasidiagonal, C00, (BCP)-operator with
spectrum and left essential spectrum the unit disc. In the last four
sections of the paper, we investigate the possible structures of the
hyperlattice of an arbitrary algebraic operator. Contrary to existing
conjectures, Hlat(T) need not be generated by the ranges and kernels
of the powers of T in the nilpotent case. In fact, this lattice
can be infinite.

This paper is a continuation of three recent articles
concerning the structure of hyperinvariant
subspace lattices of operators on a (separable, infinite dimensional)
Hilbert space $\mathcal{H}$. We show herein, in particular, that
there exists a ``universal'' fixed block-diagonal operator $B$ on
$\mathcal{H}$ such that if $\varepsilon>0$ is given and $T$ is
an arbitrary nonalgebraic operator on $\mathcal{H}$, then there exists
a compact operator $K$ of norm less than $\varepsilon$ such that
(i) $\Hlat(T)$ is isomorphic as a complete lattice to $\Hlat(B+K)$
and (ii) $B+K$ is a quasidiagonal, $C_{00}$, (BCP)-operator with
spectrum and left essential spectrum the unit disc. In the last four
sections of the paper, we investigate the possible structures of the
hyperlattice of an arbitrary algebraic operator. Contrary to existing
conjectures, $\Hlat(T)$ need not be generated by the ranges and kernels
of the powers of $T$ in the nilpotent case. In fact, this lattice
can be infinite.

Maximum principles for subharmonic
functions in the framework of quasi-regular local semi-Dirichlet
forms admitting lower bounds are presented.
As applications, we give
weak and strong maximum principles
for (local) subsolutions of a second order elliptic
differential operator on the domain of Euclidean space under conditions on coefficients,
which partially generalize the results by Stampacchia.

Maximum principles for subharmonic
functions in the framework of quasi-regular local semi-Dirichlet
forms admitting lower bounds are presented.
As applications, we give
weak and strong maximum principles
for (local) subsolutions of a second order elliptic
differential operator on the domain of Euclidean space under conditions on coefficients,
which partially generalize the results by Stampacchia.

We observe that the small quantum product of the
generalized flag manifold G/B. is a product operation star on
H*(G/B) otimes {mathbb R}[q1, dots, ql] uniquely determined by the
facts
that: it is a deformation of the cup product on H*(G/B); it is
commutative, associative, and graded with respect to deg(qi) = 4; it
satisfies a certain relation (of degree two); and the corresponding
Dubrovin connection is flat. Previously, we proved that these
properties alone imply the presentation of the ring (H*(G/B) otimes
{mathbb R}[q1, dots, ql], star) in terms of generators and relations. In
this paper we use the above observations to give conceptually new
proofs of other fundamental results of the quantum Schubert calculus
for G/B: the quantum Chevalley formula of D. Peterson (see also
Fulton and Woodward ) and the "quantization by standard
monomials" formula of Fomin, Gelfand, and Postnikov for
G = SL(n,{mathbb C}). The main idea of the proofs is the same as in
AmarzayaGuest: from the quantum {cal D}-module of G/B one can
decode all information about the quantum cohomology of this space.

We observe that the small quantum product of the
generalized flag manifold $G/B$ is a product operation $\star$ on
$H^*(G/B)\otimes \bR[q_1,\dots, q_l]$ uniquely determined by the
facts
that: it is a deformation of the cup product on $H^*(G/B)$; it is
commutative, associative, and graded with respect to $\deg(q_i)=4$; it
satisfies a certain relation (of degree two); and the corresponding
Dubrovin connection is flat. Previously, we proved that these
properties alone imply the presentation of the ring $(H^*(G/B)\otimes
\bR[q_1,\dots, q_l],\star)$ in terms of generators and relations. In
this paper we use the above observations to give conceptually new
proofs of other fundamental results of the quantum Schubert calculus
for $G/B$: the quantum Chevalley formula of D. Peterson (see also
Fulton and Woodward ) and the ``quantization by standard
monomials" formula of Fomin, Gelfand, and Postnikov for
$G=\SL(n,\bC)$. The main idea of the proofs is the same as in
Amarzaya--Guest: from the quantum $\D$-module of $G/B$ one can
decode all information about the quantum cohomology of this space.

Let A be a unital commutative associative algebra over a field of
characteristic zero, \k a Lie algebra, and
\zf a vector space, considered as a trivial module of the Lie algebra
\gf := A otimes \kf. In this paper, we give a
description of the cohomology space H2(\gf,\zf)
in terms of easily accessible data associated with A and \kf
We also discuss the topological situation, where
A and \kf are locally convex algebras.

Let $A$ be a unital commutative associative algebra over a field of
characteristic zero, $\k$ a Lie algebra, and
$\zf$ a vector space, considered as a trivial module of the Lie algebra
$\gf := A \otimes \kf$. In this paper, we give a
description of the cohomology space $H^2(\gf,\zf)$
in terms of easily accessible data associated with $A$ and $\kf$.
We also discuss the topological situation, where
$A$ and $\kf$ are locally convex algebras.

The Kronecker modules mathbb{V}(m,h,alpha), where m is a positive integer, h is
a height function, and alpha is a K-linear functional on the
space K(X) of rational functions in one variable X over an
algebraically closed field K, are models for the family of all
torsion-free rank-2 modules that are extensions of finite-dimensional
rank-1 modules. Every such module comes with a regulating polynomial
f in K(X)[Y]. When the endomorphism algebra of mathbb{V}(m,h,alpha) is
commutative and non-trivial, the regulator f must be quadratic in
Y. If f has one repeated root in K(X), the endomorphism algebra
is the trivial extension K ltimes S for some vector space S. If
f has distinct roots in K(X), then the endomorphisms form a
structure that we call a bridge. These include the coordinate rings
of some curves. Regardless of the number of roots in the regulator,
those End mathbb{V}(m,h,alpha) that are domains have zero radical. In addition,
each semi-local End mathbb{V}(m,h,alpha) must be either a trivial extension
K ltimes S or the product K times K.

The Kronecker modules $\mathbb{V}(m,h,\alpha)$, where $m$ is a positive integer, $h$ is
a height function, and $\alpha$ is a $K$-linear functional on the
space $K(X)$ of rational functions in one variable $X$ over an
algebraically closed field $K$, are models for the family of all
torsion-free rank-2 modules that are extensions of finite-dimensional
rank-1 modules. Every such module comes with a regulating polynomial
$f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}(m,h,\alpha)$ is
commutative and non-trivial, the regulator $f$ must be quadratic in
$Y$. If $f$ has one repeated root in $K(X)$, the endomorphism algebra
is the trivial extension $K\ltimes S$ for some vector space $S$. If
$f$ has distinct roots in $K(X)$, then the endomorphisms form a
structure that we call a bridge. These include the coordinate rings
of some curves. Regardless of the number of roots in the regulator,
those $\End\mathbb{V}(m,h,\alpha)$ that are domains have zero radical. In addition,
each semi-local $\End\mathbb{V}(m,h,\alpha)$ must be either a trivial extension
$K\ltimes S$ or the product $K\times K$.

S. Stahl (Canad. J. Math. 49(1997), no. 3, 617640)
conjectured that the zeros of genus polynomial are real.
L. Liu and Y. Wang disproved this conjecture on the basis
of Example 6.7. In this note, it is pointed out
that there is an error in this example and a new generating matrix
and initial vector are provided.

S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640)
conjectured that the zeros of genus polynomial are real.
L. Liu and Y. Wang disproved this conjecture on the basis
of Example 6.7. In this note, it is pointed out
that there is an error in this example and a new generating matrix
and initial vector are provided.